In my experience, it's typically best to try and prove a theorem in whatever way gets the point across, and then going back to clean up the proof, make it more concise, and think about different ways in which it could be proved.
I don't think that in general every theorem/proposition/corollary/etc can easily be approached with the hope of coming up with two different proofs; there will usually exist two (or more) substantially different proofs, but this doesn't necessarily mean that they will be at all apparent, and just understanding one proof is generally good enough for some of the higher level proofs.
To get better at proofs, it may be better to just try and come up with a single proof for each of a large collection of results, rather than try to come up with a larger number of proofs for each result. It might be better to take a middle ground, however: try and prove a large number of statements, and when you feel that another methodology could be used to prove a statement, try and prove it that way as well.
Overall, the main idea is to try and find a lot of different methodologies, but also in how to approach a problem and begin the path to writing a correct proof. The first is done using the methodology you mentioned, but can also be done by doing a lot of proofs in general, whereas for the second, doing a lot of proofs in general is probably the best approach to doing this.